L11a294

L11a293

L11a295

(Knotscape image)	See the full Thistlethwaite Link Table (up to 11 crossings). Visit L11a294 at Knotilus!

Link Presentations

[edit Notes on L11a294's Link Presentations]

Planar diagram presentation	X10,1,11,2 X12,3,13,4 X22,20,9,19 X18,7,19,8 X16,5,17,6 X4,15,5,16 X6,17,7,18 X14,22,15,21 X20,14,21,13 X2,9,3,10 X8,11,1,12
Gauss code	{1, -10, 2, -6, 5, -7, 4, -11}, {10, -1, 11, -2, 9, -8, 6, -5, 7, -4, 3, -9, 8, -3}

A Braid Representative
A Morse Link Presentation

Polynomial invariants

Multivariable Alexander Polynomial (in $u$, $v$, $w$, ...)	$-{\frac {t(1)^{2}t(2)^{5}+t(1)^{3}t(2)^{4}-3t(1)^{2}t(2)^{4}+2t(1)t(2)^{4}-2t(1)^{3}t(2)^{3}+5t(1)^{2}t(2)^{3}-4t(1)t(2)^{3}+2t(2)^{3}+2t(1)^{3}t(2)^{2}-4t(1)^{2}t(2)^{2}+5t(1)t(2)^{2}-2t(2)^{2}+2t(1)^{2}t(2)-3t(1)t(2)+t(2)+t(1)}{t(1)^{3/2}t(2)^{5/2}}}$ (db)
Jones polynomial	$q^{3/2}-3{\sqrt {q}}+{\frac {5}{\sqrt {q}}}-{\frac {8}{q^{3/2}}}+{\frac {11}{q^{5/2}}}-{\frac {13}{q^{7/2}}}+{\frac {12}{q^{9/2}}}-{\frac {11}{q^{11/2}}}+{\frac {8}{q^{13/2}}}-{\frac {5}{q^{15/2}}}+{\frac {2}{q^{17/2}}}-{\frac {1}{q^{19/2}}}$ (db)
Signature	-3 (db)
HOMFLY-PT polynomial	$z^{5}a^{7}+4z^{3}a^{7}+5za^{7}+2a^{7}z^{-1}-z^{7}a^{5}-5z^{5}a^{5}-10z^{3}a^{5}-10za^{5}-3a^{5}z^{-1}-z^{7}a^{3}-4z^{5}a^{3}-4z^{3}a^{3}+a^{3}z^{-1}+z^{5}a+3z^{3}a+za$ (db)
Kauffman polynomial	$-z^{5}a^{11}+3z^{3}a^{11}-2za^{11}-2z^{6}a^{10}+4z^{4}a^{10}-z^{2}a^{10}-3z^{7}a^{9}+6z^{5}a^{9}-4z^{3}a^{9}+2za^{9}-3z^{8}a^{8}+5z^{6}a^{8}-4z^{4}a^{8}+2z^{2}a^{8}-3z^{9}a^{7}+9z^{7}a^{7}-20z^{5}a^{7}+21z^{3}a^{7}-10za^{7}+2a^{7}z^{-1}-z^{10}a^{6}-3z^{8}a^{6}+16z^{6}a^{6}-30z^{4}a^{6}+18z^{2}a^{6}-3a^{6}-6z^{9}a^{5}+22z^{7}a^{5}-39z^{5}a^{5}+34z^{3}a^{5}-16za^{5}+3a^{5}z^{-1}-z^{10}a^{4}-4z^{8}a^{4}+23z^{6}a^{4}-36z^{4}a^{4}+20z^{2}a^{4}-3a^{4}-3z^{9}a^{3}+7z^{7}a^{3}-2z^{5}a^{3}-z^{3}a^{3}-za^{3}+a^{3}z^{-1}-4z^{8}a^{2}+13z^{6}a^{2}-11z^{4}a^{2}+4z^{2}a^{2}-a^{2}-3z^{7}a+10z^{5}a-7z^{3}a+za-z^{6}+3z^{4}-z^{2}$ (db)

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$).

\   r    \    j   \ -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 3 χ 4                       1 -1 2                     2   2 0                   3 1   -2 -2                 5 2     3 -4               7 4       -3 -6             6 4         2 -8           6 7           1 -10         5 6             -1 -12       3 6               3 -14     2 5                 -3 -16     3                   3 -18 1 2                     -1 -20 1                       1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-4$ $i=-2$ $r=-8$ ${\mathbb {Z} }$ ${\mathbb {Z} }$ $r=-7$ ${\mathbb {Z} }^{2}$ $r=-6$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-5$ ${\mathbb {Z} }^{5}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-4$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{5}$ ${\mathbb {Z} }^{5}$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{5}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Link Page master template (intermediate). See/edit the Link_Splice_Base (expert). Back to the top.

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